1. (40 points) In this problem we investigate the limits of the Pumping Lemma as it was stated in class and look for an alternative that remedies one of these shortcomings. (a) (10 points) Let L be the language L = {a'bi0 and p is a prime}. Prove that the language L = b* UL satisfies the conditions of the Pumping Lemma. I.e. show that there exists a q = N such that for every word w L with wg we can write w= xyz such that |xy| q, y| > 0, and for every i 0, xyiz L2. (b) (20 points) Prove the following generalization of the Pumping Lemma: Let L be a regular language. There exists a q = N such that for every w L and every partition of w into w = xyz with |y|9 there are strings a, b, c such that y=abc, |b| > 0, and for all i0, xabcz L. (c) (10 points) Prove that the language L2 is not regular.